The design of microfibers with mushroom-shaped tips for optimal adhesion

ABSTRACT

This invention identifies important geometric parameters of an adhesive microfiber with mushroom-shaped tip for improving and optimizing adhesive ability. The magnitude of pull-off stress is dependent on a wedge angle γ and the ratio of the tip radius to the stalk radius β of the mushroom-shaped fiber. Pull-off stress is also found to depend on a dimensionless parameter x, the ratio of the fiber radius to a length-scale related to the dominance of adhesive stress. Finally, the shape of edge tip, where the surface and sides of the mushroom-shaped tip join, is a factor that impacts strength of adhesion. Optimizing ranges for these parameters are identified.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application is a 35 U.S.C. 371 US national phase applicationof PCT international application serial number PCT/US2014/015511, filedon Feb. 10, 2014, and is herein incorporated by reference in itsentirety.

STATEMENT REGARDING FEDERALLY-SPONSORED RESEARCH AND DEVELOPMENT

This invention was made with partial government support by the NationalScience Foundation (STIR Phase I Award Number 0930610 and IPP AwardNumber 0930610). The government has certain rights in this invention.

FIELD OF THE INVENTION

This invention relates to optimizing the geometric parameters ofmicrofiber-based adhesives with mushroom tips to enhance the adhesiveperformance of the microstructure.

BACKGROUND OF THE INVENTION

We need to look no further than nature to find inspiration for many ofthe technologies we work on today. One such field that observations onnatural systems have impacted significantly in the recent years isadhesive technologies. While conventional adhesives rely on very softmaterials or viscous liquids, nature offers a unique system composed ofadhesive elements made of relatively rigid materials. These adhesiveelements are comprised of millions of tiny fibers varying in size andgeometrical complexity depending on the animal that bears them. Someinsects, spiders, and anoles have fibers with effective diameters in theorder of microns. Other animals such as the gecko lizard bearmicro-scale stalks which branch down to nano-scale fibers formingintricate hierarchical structures. The common aspect of fibrillarstructuring is its ability to conform to the adhering surface, improvecontact area and create an attractive force between individual fibersand the surface. The cumulative effect from the adhesion contribution ofevery fiber in contact is capable of generating adhesive strengths up to100 kPa as measured in the case of the gecko lizard.

A great deal of research has been performed to analyze the structure ofnatural fibrillar adhesives and measure their performance, understandthe main principles of enhanced adhesion, and fabricate syntheticcounterparts of biological fiber adhesives.

A common aspect of natural fibers among species, which is of interest inthis work, is that the cross section of a natural fiber is rarelyconstant along its length. It gradually increases close to its terminalend forming what is referred to as mushroom-shaped fibers. While initialfabrication attempts for synthetic adhesives were limited to constantcross section cylindrical fibers, realization of the actual shape ofnatural fibers has led to synthetic fibers with mushroom tips. Adhesivescomprised of mushroom-shaped fibers have shown significant improvementsover cylindrical fibers. Furthermore, measured adhesive strengths havematched and in some instances such as smooth surface applicationssurpassed adhesive strengths recorded for gecko footpads.

The force required to detach a mushroom-shaped fiber is greater thanthat of a cylindrical fiber because the contact area for amushroom-shaped fiber is larger. Work by del Campo et al., Del Campo A.,Greiner C., Arzt E. Langmuir 2007, 23, 10235-10243, reports enhancementsin pull-off loads as much as 40-fold with mushroom-shaped fibers overcylindrical fibers of equal height and stalk radius. Interestingly, formushroom-shaped fibers which exhibited this enhancement, the contactarea is only 1.7 times the contact area of flat tip cylindrical fibers.This fact points to the existence of an adhesion enhancement mechanismother than just the increase in contact area with mushroom-shapedfibers.

In this invention, the pull-off stress of mushroom-shaped fibers using acohesive zone model and finite elements (FE) simulations is examined.This model is then used to determine the optimal parameters for maximumpull-off stress. Two parameters are identified for design andoptimization: the edge angle of the fiber tip γ and the ratio of theradius of the tip to the radius of the stalk β. In addition, the impactof the shape of the edge tip—where the surface and sides of the mushroomtip intersect—is evaluated and a preferred shape is identified.

While previous literature has described that microfibers withmushroom-shaped tips demonstrate enhanced adhesion when compared withcylindrical microfibers, there exists no understanding of what geometricparameters result in microfibers with optimized adhesion.

SUMMARY OF THE INVENTION

This present invention describes a set of geometric parameters tooptimize the adhesion of mushroom-shaped microfibers. These parametersare obtained and verified using modeling and analytical approaches.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates a two-dimensional axial symmetry model for oneembodiment of a mushroom-shaped dry adhesive fiber according to thepresent invention

FIG. 2 is a scanning electron microscope image of a microfiber array ofsynthetic polyurethane mushroom-shaped fibers of the present inventionwith 4 μm stalk radius, 8 μm tip radius, and 20 μm length (Scale bar: 50μm).

FIGS. 3( a) and 3(b) are graphs of the average tensile stress (y axis)at the tip of the mushroom-shaped fiber of the present invention withβ=1.2 and γ=25° (dark gray line), γ=45° (light gray line), and γ=75°(black line), as a function of (a) normalized far field displacement Δ/h(x axis in FIG. 5( a)), and (b) normalized maximum interfacialseparation δ_(m)/δ_(c) (x axis in FIG. 5( b)). Here, δ_(c)=1 nm (χ=6).Peaks in each plot for specific θ coincide and correspond to normalizedpull-off stress Φ. Tensile load drops immediately after the maximuminterfacial separation reaches the critical separation indicating thatthe contact is unstable following crack initiation. The discontinuity atδ_(m)/δ_(c)˜0.1 prior to crack initiation in (b) marks the instant whena cohesive zone starts to form.

FIGS. 4( a)-4(d) are normalized pull-off stress Φ contour plots for x=5(FIG. 4( a)), x=10 (FIG. 4( b)), x=20 (FIG. 4( c)), and x=40 (FIG. 4(d)) as a function of β (y axis) and γ (x axis). The peak Φ for each caselies within β=1.1-1.2 and γ=45°.

FIGS. 5( a) and 5(b) are graphs of normalized pull-off stress (y axis)at the tip of a mushroom-shaped fiber of the present invention withβ=1.1 and γ={25°, 45°, 60°, 80° } as a function of (a) x (x axis), and(b) r/a (x axis) for Δ/h=0.0217. A cohesive zone is present at the edgeboth for γ=60° and γ=80° while it has not formed for γ=25° and γ=45°yet.

FIG. 6( a) is a graph of simulation results (triangle markers) ofnormalized pull-off stress (y axis) at the tip of a mushroom-shapedfiber of the present invention for β=1.2, and γ=80° as a function of x(x axis), where a crack initiates at the edge for all x. Solid linerepresents equation (2a) with B_(e)=0.84 and a=0.36. Also included arethe simulation results (circle markers) for β=1.2 and γ=45° where acrack initiates at the center for all x. Dashed line represents equation(2b) with B_(c)=0.076 and Γ_(c)=0.85.

FIG. 6( b) is a graph of simulation results (triangle markers) ofnormalized pull-off stress (y axis) at the tip of a mushroom-shapedfiber of the present invention for β=1.4, and γ=60° as a function of x(x axis), where a crack initiates at the edge for x>7 and at the centerfor x<7. Solid line represents equation (2a) with B_(e)=1.2 and a=0.21.Dashed line represents equation (2b) with a=0.17 and Γ_(c)=0.68.

FIG. 7( a) is an Illustration of three different edge angles for themushroom-tipped fibers.

FIG. 7( b) is a graph of simulation results of normalized pull-offstress (y axis) at the tip of a mushroom-shaped fiber of the presentinvention for β=1.2, and γ=45° as a function of x (x axis) for 45° wedge(diamond markers), 90° wedge (square markers), and rounded wedge(circular markers).

DETAILED DESCRIPTION OF THE INVENTION

This invention involves a dry adhesive structure and, more specifically,geometric parameters of dry adhesive microfibers with mushroom-shapedtips shape to enhance and optimize the adhesive performance of thefiber.

The application refers to the following terms, words, and phrases thathave particular meaning with regards to the present invention. Ageometric feature being micro or microscale means that at least one ofthe characteristic lengths of the feature in any 3D direction should bebetween 0.5-500 micrometers in length. Microfibers are microscale hairswith at least one microscale feature on them.

Referring generally to FIG. 1, a two-dimensional axial symmetry modelfor one embodiment of the mushroom-shaped fiber 10 of the presentinvention is shown. Fiber 10 includes stem 11 and mushroom-shaped tip 12with a flat surface 13. Stem 11 and tip 12 are symmetrical aboutsymmetry axis 14, such that radius a of stem 11 (up to the point ofconnection with tip 12, in a preferred embodiment, is constant along theheight of stem 11, and radius a_(t) of tip 12, in a preferredembodiment, is constant at surface 13 as well and is fixed in radialdirection to simulate full-friction contact. FIG. 1 also shows height h,which is the distance between the bottom 15 of stem 11 and surface 13.Thus, in a preferred embodiment, surface 13 and the cross-section ofstem 11 are circular. In other embodiments, however, an oval orelliptical shape and/or cross-section may be used. The shape of theinclined sides on the underside of the mushroom tip is linear is linearin a preferred embodiment, but can be alternately convex or concave withrespect to the stem axial direction and tip surface.

In operation and use, fiber 10 attaches, adheres or otherwise is joinedat bottom 15 to a base or substrate (not shown). Fiber 10 issubstantially perpendicular to the base or substrate. As is known in theart, surface 13 has adhesive forces that allow fiber 10 to attach and/oradhere to other objects. The present invention addresses and describesoptimum parameters of fiber 10 for maximizing the adhesive force atsurface 13. These parameters focus, referring to FIG. 1, on (i) wedgeangle γ, (ii) the ratio β of the radius a_(t) of the tip 12 to theradius a of the stem 11, and (iii) the sharpness or radius of curvatureof the outer edge 16 formed by the surface 13 and the inclined sides 17of the tip 12. In embodiments where a_(t) and a are not constant, theeffective radius of surface 13 and/or of the cross-section of stem 11 isused. Similarly, where surface 13 is an oval or ellipse, the radiusa_(t) is determined where a_(t) is smallest. In embodiments where sides17 are not linear, that is, they are concave or convex, wedge angle γ isdetermined by using the line that is formed from edge 16 to the apex ofthe radius of curved side 17.

The height h of fiber 10 is not a controlling geometric parameter. Thisinvention may be applied to microfibers or nanofibers currently incommercial use, including those with characteristic geometries rangingfrom 0.2-500 nm and 0.2-500 μm in length. As shown in FIG. 2 each fiber10 is, in one embodiment, part of a microfiber array of syntheticpolyurethane mushroom-shaped fibers of the present invention. In FIG. 2,the fibers have 4 μm stalk radius, 8 μm tip radius, and 20 μm length(Scale bar: 100 μm).

As detailed below, the pull-off stress at surface 13 is important inevaluating the adhesive forces at surface 13. In particular, pull-offstress for an individual mushroom-shaped fiber can be modelled using DBcohesive zone model and finite element (“FE”) analyses. These analysesreveal critical information about the detachment mechanism ofmushroom-shaped fibers and how this behaviour is influenced by thegeometry and interfacial properties of fiber, namely, optimal values forthe parameters β and γ, as well as the shape of the edge angle. Moredirectly, these analyses reveal geometrical guidelines to ensureoptimal, i.e., high and robust, adhesion relative to the intrinsicadhesive stress.

More specifically, the adhesion mechanism of bio-inspiredmushroom-shaped fibers is investigated by implementing theDugdale-Barenblatt cohesive zone model into FE simulations. As detailedbelow, these simulation show that the magnitude of pull-off stressdepends on the edge angle γ and the ratio of the tip radius to the stalkradius β of the mushroom-shaped fiber. Pull-off stress is also found todepend on a dimensionless parameter x, the ratio of the fiber radius toa length-scale related to the dominance of adhesive stress. In thisinvention, β=1.1-1.2 and γ=45° are described as the optimal parametersfor maximizing pull-off stress for a single mushroom-shaped fiber. Inother embodiments of the invention, β may range from 1.05 to 1.4 and γmay range from 20° to 55° for mushroom-shaped tips geometries whichdemonstrate improved adhesion with respect to a cylindrical microfiber.As further detailed below, in the discussion of crack initiation and apull-off model, the location of crack initiation is found to depend on xfor a given β and γ. An analytical model for pull-off stress, whichdepends on the location of crack initiation as well as γ and β, isproposed and found to agree with the simulation results. In the nextdiscussion section, the simulation results are also substantiated by apull-off stress comparison between mushroom-shaped and cylindricalfibers. Finally, the importance of the edge angle, as a parameter, isdiscussed.

Cohesive Zone Model.

As a starting point, adhesion problems between surface 13 and anotherobject in contact with surface 13 were studied using a cohesive zonemodel such as the Dugdale-Barenblatt (DB) model. Dugdale D. S. J MechPhys Solids 1960, 8, 100-104; Barenblatt G. I. Advances in AppliedMechanics (eds. H. L. Dryden T.v.K.G.K.F.H.v.d.D., Howarth L.) 1962, pp.55-129, Elsevier. It is a simple cohesive zone model where the interfacebetween surface 13 and the object in contact with surface 13 separateswhen the normal interfacial stress reaches the theoretical strength ofthe interface σ_(o). The interface continues to separate at σ_(o) untilthe separation reaches a critical distance δ_(c), after which theinterface can no longer support stress, resulting in the initiation of acrack between surface 13 and the object in contact with surface 13. Theregion where the separation of interface occurs is referred to as thecohesive zone. In this model, the work of adhesion is given byω_(adh)=σ_(o)δ_(c). Tang et al. found the pull-off force of a soft,elastic cylindrical fiber in contact with a rigid flat surface whoseheight is much larger than its radius using the DB cohesive zone model.Tang T., Hui C. Y., Glassmaker N. J. J R Soc Interface 2005, 2, 505-516.According to this study, normalized pull-off stress Φ≡σ_(s)/σ_(o)depends on a single dimensionless parameter x defined as:

$\begin{matrix}{{\chi \equiv \frac{\sigma_{o}^{2}{a\left( {1 - v^{2}} \right)}}{2\pi \; {Ew}_{adh}}} = {\frac{\sigma_{o}{a\left( {1 - v^{2}} \right)}}{2\pi \; E\; \delta_{c}}.}} & (1)\end{matrix}$

Here, a is the radius, E is the elastic modulus, and ν is the Poisson'sratio of the fiber—which all can be measured for a particular fiber sizeand material. Work of adhesion, ω_(adh), is also a measurable quantity.Critical separation distance, δ_(c), is derived from van der Waalsinteractions. It is the distance between two flat surfaces when theinterfacial attractive stress is the highest and it is usually assumedto be in the range 0.1 nm-1 nm. According to DB model, work of adhesionis the product of intrinsic adhesive strength and the criticalseparation distance. The intrinsic adhesive strength, σ_(o), can becalculated using work of adhesion and the critical separation distance.The magnitude of intrinsic adhesive strength is limited by the Young'smodulus (E) of the material for soft materials (materials with E up to10 MPa) due to stress blunting and thermal fluctuations.

Dimensionless parameter x is the ratio of the fiber radius to a lengthscale related to the dominance of the interfacial adhesive forces.Hence, when x<<1, an attractive stress that is equal to the intrinsicadhesive stress covers the entire fiber tip and pull-off stressapproaches the theoretical limit, Φ=1 (i.e. σ_(s)=σ_(o)). This regime isreferred to as the flaw insensitive regime. On the other hand, whenx<<1, σ_(o) acts over a small portion of the interface, which results inpull-off stress being much smaller than the theoretical limit,σ_(s)<<σ_(o). This regime is referred to as the flaw sensitive regime.The non-dimensional parameter x is also relevant to the adhesion problemof mushroom-shaped fibers. Thus, pull-off results can be presented as afunction of x, and a pull-off stress model based on x is presentedbelow.

FIGS. 3( a) and 3(b) show the results of the simulations performed for amushroom-shaped fiber illustrated in FIG. 1 using COMSOL MultiPhysics4.3 FE analysis software. In these simulations, it is assumed that thesurface 13 of fiber 10 is in full friction contact with a rigid smoothsurface of another object. The DB cohesive zone model was implemented atthe tip of the fiber in FE simulations and modified slightly to avoiddivergence. Step function from zero stress to σ_(o) in the cohesive zonewas replaced with a high stiffness relation between the attractivestress and interfacial opening where the interface is required toseparate by 10% of δ_(c) before cohesive zone forms (i.e., before σ_(o)is reached). Aksak B., Hui C. Y., Sitti M. J R Soc Interface 2011, 8,1166-1175. Unless stated otherwise, the simulation parameters are a=1μm, E=3 MPa, ν=0.5, and σ_(o)=100 kPa. In the simulation, the tip radiusa_(t) varies from 1.05 μm to 2 μm while the edge angle γ varies from 25°to 80°. It is important to note that the tip edge or corner 16 forms awedge with a wedge angle that is also equal to γ. The effect of wedgeangle of edge 16 being different from γ is addressed below in thediscussion of the impact of the edge angle. The height of the fiber h isfixed at 10 μm. Dimensionless parameter x is varied by changing δ_(c)for fixed a_(t), E, ν, and σ_(o). While applying displacement Δgradually to the base of the fiber, pull-off load is determined from thefar field tensile stress σ_(ff) the fiber bears when the maximuminterfacial separation equals δ_(c). The tensile load reaches itsmaximum at the instant δ_(c) is reached at the interface as shown inFIG. 3( a). Knowing pull-off load p_(s), pull-off stress is calculatedfrom σ_(s)=p_(s)/(πa_(t) ²).

Peaks in each plot for specific γ coincide and correspond to normalizedpull-off stress Φ. Tensile load drops immediately after the maximuminterfacial separation reaches the critical separation indicating thatthe contact is unstable following crack initiation. As shown in FIG. 3(b), the discontinuity at δ_(m)/δ_(c)˜0.1 prior to crack initiation marksthe instant when a cohesive zone starts to form.

Pull-off stress for all tip-to-base ratios β≡a_(t)/a and edge angles γare shown for select x values in FIGS. 4( a)-(d). Contour plots showclear peaks at β=1.1-1.2 and γ=45° suggesting that these values areoptimal for maximum pull-off stress. In addition, the peak pull-offstress drops with increasing x with highest value at Φ=0.97 for x=5 andlowest value at Φ=0.88 for x=40 both obtained for β=1.1 and γ=45°.

FIG. 5( a) shows pull-off stress for β=1.1 for all γ as a function of x.In line with the data presented in FIGS. 4( a)-(d), γ=45° yields thehighest pull-off stress for all x, except when x<<1 (shown at the leftof FIG. 5( a). In the regime where x<<1, pull-off is flaw insensitive,and it is expected that Φ=1 regardless of β, γ, and x.

FIG. 5( a) further shows that for γ≦45°, pull-off stress saturatestowards a constant value as x increases. In contrast for γ>45°, pull-offstress continuously drops with increasing x. The dependence of pull-offstress on the wedge angle at the limit χ→∝ (i.e. δ_(c)→0) is reinforcedusing Bogy's study on stress singularities at bimaterial wedgeinterfaces. Carbone G., Pierro E., Gorb S. N. Soft Matter 2011, 7,5545-5552. For a soft incompressible elastomer (i.e., Poisson's ratio,ν=0.5) in contact with a rigid substrate, stress at the edge of thefiber tip is finite for γ≦45° and singular for γ>45°. Stress profiles ofa mushroom-shaped fiber with β=1.1 and γ={25°, 45°, 60°, 80° } are shownfor a far field displacement of Δ/h=0.0217 in FIG. 5( b). Due to thesingularity, a cohesive zone is present at the edge of the fiber bothfor γ=60° and γ=80° while cohesive zone has not formed, as yet, forγ=25° and γ=45°. Assuming that δ_(c)=0, this implies that pull-off willoccur shortly after the maximum tensile stress at the interface reachesσ_(o). If there is a stress singularity at the edge of the tip, normalstress will be equal to σ_(o) at the edge the moment a tensile load isapplied to the fiber. The interface will open starting at the edge andpull-off will depend on whether this opening is less than or equal toδ_(c). On the other hand, if stress is finite everywhere at theinterface, a sufficiently large tensile load has to be applied before acohesive zone starts forming. This implies that regardless of the valueof δ_(c), pull-off load has a finite lower limit when γ≦45°. Theexistence of this lower limit provides robust adhesion becauseregardless of the value of x, one can expect to obtain pull-off stressequal to at least this lower. In particular for fibers with γ=45° andβ=1.1-1.2, this lower limit for normalized pull-off stress is remarkablyΦ=0.85 which ensures high and robust adhesion. Stated differently, theseresults demonstrate that, no matter what the interfacial values may be,85% of intrinsic adhesive strength can be recovered if there is notsingularity present. Thus confirming that optimum adhesion can beobtained at γ=45°.

FIGS. 4( a)-(d) and 5(a) and (b), together, show preferred values for γand β when x ranges from 1 to 100. In particular, within this range forx, normalized pull-off stress of 0.7-0.9 is observed where the ratio ofa_(t)/a is 1.05-1.4, and wedge angle γ is 20-55. In the same range,pull-off stress of 0.8-0.9 is observed where the ratio of a_(t)/a is1.1-1.2, and wedge angle γ is 40-50. Finally, an optimum of pull-offstress of 0.9 or greater is obtained where the ratio of a_(t)/a is1.1-1.2, and wedge angle γ is 45.

Crack Initiation and Pull-Off Model.

The location of crack initiation (i.e. where the interface openingequals δ_(c)) is also dependent both on tip shape parameters γ and β,and the value of x. For γ≦45°, since normal stress is finite everywhereat the tip, the cohesive zone forms when and where the maximum normalstress reaches σ_(o). This location corresponds to r/a=0 (center) forγ=45° and all β [refer to FIG. 5( b) for normal stress profiles].Simulation results show that the location of crack initiation isindependent of x for this set of tip parameters. For γ>45° on the otherhand, the crack initiation is always at the edge for sufficiently largex. For certain combinations of β and γ, for instance β=1.4 and γ=60°,although the normal stress at the tip is highest at the edge, it doesnot increase monotonically from the center to the edge of the tip.Stress profile has a minimum at r/a˜1 [FIG. 5( b)]. For sufficiently lowx, gradually increasing far field displacement causes a cohesive zone toform at the edge first. Since δ_(c) is relatively large for low x, anincrease in tension does not immediately result in a crack to initiateat the edge. In the meantime, the stress in the center graduallyincreases reaching σ_(o) where a second cohesive zone starts to form.The center separates faster than the edge which results in a crack toinitiate at the center. The simulations, therefore, lead to theconclusion that crack initiation switches to the center of the fiber ifstress at the center is able to reach σ_(o) prior to a crack initiatingat the edge.

Similar to the pull-off stress model proposed by Tang et al., as derivedwith the assumption that the size of the cohesive zone is much smallercompared to the tip radius, pull-off stress can be estimated accordingto where the crack initiates as:

$\begin{matrix}{\Phi = {\frac{\sigma_{s}}{\sigma_{o}}\left\{ {\begin{matrix}{B_{e}\left\lbrack {\beta \; \chi} \right\rbrack}^{- \alpha} & {{{for}\mspace{14mu} {edge}\mspace{14mu} {crack}},} \\{{B_{c}\lbrack{\beta\chi}\rbrack}^{- 0.5} + \Gamma_{c}} & {{for}\mspace{14mu} {center}\mspace{14mu} {crack}}\end{matrix},} \right.}} & \begin{matrix}\left( {2a} \right) \\\left( {2b} \right)\end{matrix}\end{matrix}$

For a crack initiating at the center, there is a square-root singularityand thus a=0.5 as shown in equation (2b). Numerical constants B_(e) andB_(e) are form factors which are determined by fitting equation (2a) andequation (2b) to the simulation data for given β, γ, and x. ConstantΓ_(c) is approximately the value of the pull-off stress when the crackinitiation switches from the edge to the center of the fiber for γ>45°.Constant Γ_(e) is determined by fitting equation (2b) to simulationdata. For γ≦45° and a crack initiating at the center for all x, Γ_(c) isthe lower bound for normalized pull-off stress at the limit x→∞. FIG. 6shows the simulation data and the model fits using equation (2a) andequation (2b). The proposed model is in agreement with the simulationdata except when x→0. Equations 2(a) and (b) are not valid in thisregime as the cohesive zone occupies a relatively larger portion of thetip. Additionally, for certain combinations of β and γ, a crackinitiates at r/a˜1 due to a stress peak at this location. Equations 2(a)and (b) also are not valid in this case.

Crack initiation at the center of the fiber shown in simulations wasalso seen experimentally by Varenberg et al., Varenberg M., Gorb S. J RSoc Interface 2008, 5, 785-789, and Murphy et al., Murphy M. P., AksakB., Sitti M. Small 2009, 5, 170-175. Varenberg et al. presentedexperimental results with mushroom-shaped fibers using high speedimaging and showed that the detachment of the fiber was initiated withan internal crack. Similarly, Murphy et al. observed under tensileloading of polyurethane mushroom-shaped fibers that an internal crackformed and propagated in a matter of milliseconds leading to contactfailure.

These simulation results all suggest that in the flaw insensitive regimewhere x<<1, regardless of the value of γ and β, Φ=1. Further, when xranges from 1-100, these results show preferred values of γ and β tooptimize adhesion.

Pull-Off Stress Comparison Between Mushroom-Shaped and CylindricalFibers.

For x<<1, i.e., for very small fiber radius (typically less than 1 μm),pull-off stress equals intrinsic adhesive stress and is insensitive tothe tip shape. This is in agreement with the simulation results outlinedabove. However, for large x, tip shape significantly affects adhesion.del Campo et al. measured pull-off loads for mushroom-shaped fibers andcylindrical fibers of the same height; stalk radius and packing density.While this study measured approximately 0.7 mN for cylindrical fiberswith a hemispherical glass indenter, the measured pull-off load wasapproximately 28 mN with mushroom-shaped fibers, an enhancement of40-fold. Reported values are approximate pull-off loads near saturationas interpreted from the graphical data presented in del Campo et al. Anenhancement factor e can be defined as the ratio of the pull-off loadbetween two different fiber arrays. For an experiment that useshemispherical glass indenter with a radius much larger than thedimensions of an individual fiber in the array, the enhancement factorof mushroom-shaped fibers over the cylindrical fibers with the samepacking density, stalk radius, height, and material becomes:

$\begin{matrix}{e = {\frac{P_{s,m}}{P_{s,c}} = {\beta^{4}\left( \frac{\sigma_{s,m}}{\sigma_{s,c}} \right)}^{2}}} & (3)\end{matrix}$

Here, P_(m) and P_(c) are the pull-off loads for the mushroom-shaped andcylindrical fiber arrays, respectively. The del Campo et al experimentswere carried out with polydimethylsiloxane (PDMS) cylindrical andmushroom-shaped fibers. For cylindrical fibers, a=10 μm and h=25 μm. Thetip radius for mushroom-shaped fibers were a_(t)=13 μm and they sharedthe same a and h with the cylindrical fibers. For both samples, thepacking densities (number of fibers per unit area) were the same.According to the model of the present invention, β=1.3 and although theedge angle was not reported in the del Campo et al work, γ=45° isassumed for simplicity. For β=1.3 and γ=45°, a conservative estimate ofnormalized pull-off stress according to the model of the presentinvention is σ_(s,m)=Γ_(c)σ_(c)=0.74σ_(o). For cylindrical fibers,σ_(s,c)=0.83x^(−0.4)σ_(o) as shown by Tang et al. For the describedfiber arrays, equation (3) becomes e=2.27χ^(0.8). If PDMS is assumed tobe incompressible (ν=0.5) with elastic modulus E=1.42 MPa, forglass-PDMS contact, ω_(adh)=25 mJ/m². If an intrinsic adhesive stress ofσ_(o)=1 MPa is also assumed (a reasonable value for PDMS), plugging inthe material properties, interfacial properties and fiber dimensionsinto equation (1), one finds x=33.6, and, in turn, e=38. This estimateis close to e=40 that del Campo et al. obtained in their measurements.

Implication of Edge Angle.

The manufacturing technique used to fabricate mushroom-shaped fibers maynot yield a sharp edge or corner (i.e. wedge) for individual fibers atthe edge 16 of the tip. This implies that the wedge angle will bedifferent from the angle of edge 16 defined in this work which maysignificantly affect the pull-off stress. To demonstrate this effect,simulations for mushroom-shaped fibers were performed with β=1.2 andγ=45° employing three different edge shapes; namely 45° edge, 90° edgeand rounded edge, as shown in FIG. 7( a). The radius of curvature wasset to 10 nm for the rounded edge. In case of the 90° edge, a 10 nm highrectangle was added to the tip. For all three cases, the size of the tipin contact was kept at 1.2 μm ensuring constant β=1.2. As shown in FIG.7( b), the shape of the edge angle has significant effect on pull-offstress for relatively high x values where a crack initiates at the edge16 for both the rounded and 90° edge. Below a critical x where a crackinitiation transitions to the center, pull-off stress for all threecases follow the same path once a crack is initiated at the centerindicating little dependence on edge angle. Thus, a qualitative argumentsuggests that while fibers with small diameters and high elastic modulus(large x) favor edge angle independence, the dependence of pull-offstress to the edge angle is less significant for high strengthinterfaces (small x). This is assuming that the critical separationdistance is in the order of 1 nm and somewhat constant for van der Waalsinteractions.

In practical application, the fiber material and manufacturing processprovide limitations on the ability to obtain a true sharp angle for edge16. Wetting properties also impact this process, with wetting allowingfor sharper edges. In a preferred embodiment, wetted polymer materialsare used to maximize the sharpness of the edge 16. However, materialsand manufacturing limitations will still provide some radius ofcurvature. A preferred radius of curvature for maximum sharpness in thispreferred embodiment is less than or equal to 10 nm. For othermaterials, maximum sharpness, with the smallest radius of curvaturepossible—given the material, wetting properties and manufacturingprocess—is preferred in order to optimize the edge shape parameter. Inthese cases, microfibers with a sharpness of the edge 16 with radius ofcurvature of 0.1 nm up to 10 μm will still demonstrate stronger adhesionwhen compared with microfibers with larger edge 16 radius of curvature.

In summary, pull-off stress for an individual mushroom-shaped fiber wasmodelled using DB cohesive zone model and FE analyses. This studyrevealed critical information about the detachment mechanism ofmushroom-shaped fibers and how this behaviour is influenced by thegeometry as well as the interfacial properties. A simple geometricalguideline to ensure high and robust adhesion relative to the intrinsicadhesive stress was offered. While these results are important fordesigning dry fibrillar adhesives, they are only concerned with theperformance when the loading is in the axial direction of fibers. Theeffect of shear loading should also be considered along with the resultsof this study in designing fibrillar adhesives.

This invention may be applied to design mushroom-tipped microfibers withoptimized tip geometry for maximum pull-off adhesion. Individualmushroom-tipped microfibers may be incorporated into arrays ofmicrofibers to produce surfaces, such as two dimensional flexible orrigid tapes or the flat or curved surfaces of molded three dimensionalobjects, with high adhesion to a broad range of substrates. Thesematerials with high adhesion surfaces have value across a broad range ofindustries and markets. This invention will add value to anymicrofiber-based adhesive with mushroom tip shape by increasing theadhesive ability of each microfiber, resulting in a product capable ofstronger adhesion per unit area. This increase in adhesion will enableaccess to new product markets where stronger interfacial adhesionstrength is required. Additionally, it will reduce the surface area ofthe product required to meet a target adhesion specification, resultingin a more cost-effective solution as well as minimizing the packaging ofthe product, which may additionally open up new product applicationswith space constraints.

More specifically, this invention may be applied to designmushroom-shaped tip microfibers with optimized tip geometry for maximumpull-off adhesion. Individual mushroom-tipped microfibers may beincorporated into arrays of microfibers to produce surfaces, such as twodimensional flexible or rigid tapes or the flat or curved surfaces ofmolded three dimensional objects, with high adhesion to a broad range ofsubstrates. These materials with high adhesion surfaces have valueacross a broad range of industries and markets. This invention will addvalue to any microfiber-based adhesive with a mushroom tip shape byincreasing the adhesive ability of each microfiber, resulting in aproduct capable of stronger adhesion per unit area. This increase inadhesion will enable access to new product markets where strongerinterfacial adhesion strength is required. Moreover, it will reduce thesurface area of the optimized mushroom-tipped microfiber array requiredto meet a target adhesion specification (when compared with a suboptimalmushroom-tipped microfiber array). This means that consumers ofmushroom-tipped-microfiber arrays which make use of the invention willneed fewer units of the adhesive product to accomplish the same adhesiveobjective when compared with suboptimal product. The resultingperformance enhancement made possible by the invention reduces the inputmaterial required to manufacture each unit of adhesion, and thus thecost of producing mushroom-based microfiber adhesives is also reduced.These cost savings can be passed on to consumers, resulting in a productwhich can be more competitively priced than competing suboptimaladhesives. This performance optimization also results in reducedshipping costs per unit of adhesion, as well as reduced packaging costs,additionally enhancing the competitive price advantage made possible bythis invention. These optimized adhesives also allow access toapplications where space constraints are critical, where suboptimaladhesives may not be able to provide the necessary adhesive strength foran established product size

Areas of the mushroom-shaped microfibers of the present invention aremade through molding processes including vacuum-assisted or non-vacuumassisted manual or automated processes. One potential manufacturingmethod for areas of optimized mushroom-tipped microfibers isroll-to-roll manufacturing of continuous tapes. Other molding processeswhich may be used to produce these microfibers include, but are notlimited to:

A. Injection molding: Injection over molding, Co-injection molding, Gasassist injection molding, Tandem injection molding, Ram injectionmolding, Micro-injection molding, Vibration assisted molding, Multilinemolding, Counter flow molding, Gas counter flow molding, Melt counterflow molding, Structural foam molding, Injection-compression molding,Oscillatory molding of optical compact disks, Continuous injectionmolding, Reaction injection molding (Liquid injection molding, Solublecore molding, Insert molding), and Vacuum Molding;B. Compression molding: Transfer molding, and Insert molding;C. Thermoforming: Pressure forming, Laminated sheet forming, Twin sheetthermoforming, and Interdigitation;D. Casting: Encapsulation, Potting, and impregnation;E. Coating Processes: Spray coating, Powder coatings, Vacuum coatings,Microencapsulation coatings, Electrode position coatings, Floc coatings,and Dip coating;F. Blow molding: Injection blow molding, Stretch blow molding, andExtrusion blow molding;G. Vinyl Dispersions: Dip molding, Dip coatings, Slush molding, Spraycoatings, Screened inks, and Hot melts; andH. Composite manufacturing techniques involving molds: Autoclaveprocessing, Bag molding, Hand lay-up, and Matched metal compression.

In one preferred embodiment of the invention, the microfibers areproduced from polyurethane having a 60 Shore A hardness. In otherembodiments, the microfibers may have a hardness ranging from 10 Shore Ato 100 Shore D. In other embodiments of the invention, themushroom-tipped microfibers with optimized tip geometry can be producedfrom any moldable plastic, including:

A. Thermosets:

i. Formaldehyde Resins (PF, RF, CF, XF, FF, MF, UF, MUF);ii. Polyurethanes (PU);iii. Unsaturated Polyester Resins (UP);iv. Vinylester Resins (VE), Phenacrylate Resins, Vinylester Urethanes(VU);v. Epoxy Resins (EP);vi. Diallyl Phthalate Resins, Allyl Esters (PDAP);vii. Silicone Resins (Si); andviii. Rubbers: R-Rubbers (NR, IR, BR, CR, SBR, NBR, NCR, IIR, PNR, SIR,TOR, HNBR), M-Rubbers (EPM, EPDM, AECM, EAM, CSM, CM, ACM, ABM, ANM,FKM, FPM, FFKM), O-Rubbers (CO, ECO, ETER, PO), Q-(Silicone) Rubber (MQ,MPQ, MVQ, PVMQ, MFQ, MVFQ), T-Rubber (TM, ET, TCF), U-Rubbers (AFMU, EU,AU) Text, and Polyphosphazenes (PNF, FZ, PZ)

B. Thermoplastics

i. Polyolefins (PO), Polyolefin Derivates, and Copoplymers: StandardPolyethylene Homo- and Copolymers (PE-LD, PE-HD, PE-HD-HMW, PE-HD-UHMW,PE-LLD); Polyethylene Derivates (PE-X, PE+PSAC); Chlorinated andChloro-Sulfonated PE (PE-C, CSM); Ethylene Copolymers (ULDPE, EVAC,EVAL, EEAK, EB, EBA, EMA, EAA, E/P, EIM, COC, ECB, ETFE; PolypropyleneHomopolymers (PP, H-PP)ii. Polypropylene Copoplymers and -Derivates, Blends (PP-C, PP-B, EPDM,PP+EPDM)iii. Polybutene (PB, PIB)iv. Higher Poly-a-Olefins (PMP, PDCPD)v. Styrene Polymers: Polystyrene, Homopolymers (PS, PMS); Polystyrene,Copoplymers, Blends; Polystyrene Foams (PS-E, XPS)vi. Vinyl Polymers: Rigid Polyvinylchloride Homopolymers (PVC-U);Plasticized (Soft) Polyvinylchloride (PVC-P); Polyvinylchloride:Copolymers and Blends; Polyvinylchloride: Pastes, Plastisols,Organosols; Vinyl Polymers, other Homo- and Copolymers (PVDC, PVAC,PVAL, PVME, PVB, PVK, PVP)vii. Fluoropolymers: FluoroHomopolymers (PTFE, PVDF, PVF, PCTFE); FluoroCopolymers and Elastomers (ECTFE, ETFE, FEP, TFEP, PFA, PTFEAF,TFEHFPVDF (THV), [FKM, FPM, FFKM])viii. Polyacryl- and Methacryl Copolymersix. Polyacrylate, Homo- and Copolymers (PAA, PAN, PMA, ANBA, ANMA)x. Polymethacrylates, Homo- and Copolymers (PMMA, AMMA, MABS, MBS)xi. Polymethacrylate, Modifications and Blends (PMMI, PMMA-HI, MMA-EMLCopolymers, PMMA+ABS Blendsxii. Polyoxymethylene, Polyacetal Resins, Polyformaldehyde (POM):Polyoxymethylene Homo- and Copolymers (POM-H, POM-Cop.);Polyoxymethylene, Modifications and Blends (POM+PUR)xiii. Polyamides (PA): Polyamide Homopolymers (AB and AA/BB Polymers)(PA6, 11, 12, 46, 66, 69, 610, 612, PA 7, 8, 9, 1313, 613); PolyamideCopolymers, PA 66/6, PA 6/12, PA 66/6/610 Blends (PA+: ABS, EPDM, EVA,PPS, PPE, Rubber); Polyamides, Special Polymers (PA NDT/INDT [PA 6-3-t],PAPACM 12, PA 6-I, PA MXD6 [PARA], PA 6-T, PA PDA-T, PA 6-6-T, PA 6-G,PA 12-G, TPA-EE); Cast Polyamides (PA 6-C, PA 12-C); Polyamide forReaction Injection Molding (PA-RIM); Aromatic Polyamides, Aramides(PMPI, PPTA)xiv. Aromatic (Saturated) Polyesters: Polycarbonate (PC); Polyesters ofTherephthalic Acids, Blends, Block Copolymers; Polyesters of AromaticDiols and Carboxylic Acids (PAR, PBN, PEN)xv. Aromatic Polysulfides and Polysulfones (PPS, PSU, PES, PPSU,PSU+ABS): Polyphenylene Sulfide (PPS); Polyarylsulfone (PSU, PSU+ABS,PES, PPSU)xvi. Aromatic Polyether, Polyphenylene Ether, and Blends (PPE):Polyphenylene Ether (PPE); Polyphenylene Ether Blendsxvii. Aliphatic Polyester (Polyglycols) (PEOX, PPDX, PTHF)xviii. Aromatic Polyimide (PI): Thermosetting Polyimide (PI, PBMI, PBI,PBO, and others); Thermoplastic Polyimides (PAI, PEI, PISO, PMI, PMMI,PESI, PARI);xiv. Liquid Crystalline Polymers (LCP)xv. Ladder Polymers: Two-Dimensional Polyaromates and -Heterocyclenes:Linear Polyarylenes; Poly-p-Xylylenes (Parylenes);Poly-p-Hydroxybenzoate (Ekonol); Polyimidazopyrrolone, Pyrone;Polycyclonexvi. Biopolymers, Naturally Occurring Polymers and Derivates: Cellulose-and Starch Derivates (CA, CTA, CAP, CAB, CN, EC, MC, CMC, CH, VF, PSAC);2 Casein Polymers, Casein Formaldehyde, Artificial Horn (CS, CSF);Polylactide, Polylactic Acid (PLA); Polytriglyceride Resins (PTP®); xix.Photodegradable, Biodegradable, and Water Soluble Polymers;xvii. Conductive/Luminescent Polymers;xviii. Aliphatic Polyketones (PK);xix. Polymer Ceramics, Polysilicooxoaluminate (PSIOA);xx. Thermoplastic Elastomers (TPE): Copolyamides (TPA), Copolyester(TPC), Polyolefin Elastomers (TPO), Polystyrene Thermoplastic Elastomers(TPS), Polyurethane Elastomers (TPU), Polyolefin Blends with CrosslinkedRubber (TPV), and Other TPE, TPZ; andxxi. Other materials known to those familiar with the art.

While the disclosure has been described in detail and with reference tospecific embodiments thereof, it will be apparent to one skilled in theart that various changes and modifications can be made therein withoutdeparting from the spirit and scope of the embodiments. Thus, it isintended that the present disclosure cover the modifications andvariations of this disclosure provided they come within the scope of theeventual appended claims and their equivalents.

What is claimed is:
 1. A dry adhesive fiber structure with optimizedgeometric parameters for increasing adhesive performance, comprising: adry adhesive fiber including: a mushroom-shaped tip having a flatsurface with a radius a_(t), inwardly inclined flat sides that form awedge angle γ with the surface, and an end opposite the surface; and astem with a cross-sectional area having (i) a distal end connected tothe end of the tip at a tip point of connection, (ii) a proximal endopposite the distal end, and (iii) a longitudinal length with a radius afrom the distal end to the proximal end; wherein the ratio of a_(t)/a is1.05-1.4, and wedge angle γ is 20-55.
 2. The dry adhesive fiberstructure according to claim 1, wherein ratio of a_(t)/a is 1.1-1.2, andwedge angle γ is 40-50.
 3. The dry adhesive fiber structure according toclaim 1, wherein ratio of a_(t)/a is 1.1-1.2, and wedge angle γ is 45.4. The dry adhesive fiber according to claim 1, wherein the tip edgeformed where the tip surface meets the tip inclined sides has a radiusof curvature of less than 10 μm for wetted polymer materials.
 5. The dryadhesive fiber according to claim 1, wherein the tip edge formed wherethe tip surface meets the tip inclined sides has maximum sharpness forthe fiber material and manufacturing process used to create the fiber.6. The dry adhesive fiber according to claim 1, wherein the fiber lengthvaries from 0.5 nm to 500 μm.
 7. The dry adhesive fiber according toclaim 1, wherein the shape of the inclined sides on the underside of themushroom tip is selected from the group consisting of linear, convex,and concave shapes with respect to the stem axial direction and tipsurface.
 8. A dry adhesive fiber structure with optimized geometricparameters for increasing adhesive performance, comprising: a dryadhesive fiber including: a mushroom-shaped tip having a flat surfacewith a radius a_(t), inwardly inclined flat sides that form a wedgeangle γ with the surface, and an end opposite the surface; and a stemwith a cross-sectional area having (i) a distal end connected to the endof the tip at a tip point of connection, (ii) a proximal end oppositethe distal end, and (iii) a longitudinal length with a radius a from thedistal end to the proximal end; wherein the ratio of a_(t)/a is1.05-1.4, the wedge angle γ is 20-55, and the normalized pull-off stressat the fiber surface is 0.7-0.9.
 9. The dry adhesive fiber structureaccording to claim 8, wherein ratio of a_(t)/a is 1.1-1.2, the wedgeangle γ is 40-50, and the normalized pull-off stress at the fibersurface is 0.8-0.9.
 10. The dry adhesive fiber structure according toclaim 8, wherein ratio of a_(t)/a is 1.1-1.2, the wedge angle γ is 45,and the normalized pull-off stress at the fiber surface at 0.9.
 11. Thedry adhesive fiber according to claim 8, wherein the tip edge formedwhere the tip surface meets the tip inclined sides has a radius ofcurvature of less than 10 nm for wetted polymer materials.
 12. The dryadhesive fiber according to claim 8, wherein the fiber length variesfrom 0.5 nm to 500 μm.
 13. The dry adhesive fiber according to claim 8,wherein the shape of the inclined sides on the underside of the mushroomtip is selected from the group consisting of linear, convex, and concaveshapes with respect to the stem axial direction and tip surface.
 14. Adry adhesive fiber structure with optimized geometric parameters forincreasing adhesive performance, comprising: a dry adhesive fiberincluding: a mushroom-shaped tip having a flat surface with a radiusa_(t), inwardly inclined flat sides that form a wedge angle γ with thesurface, and an end opposite the surface; and a stem with across-sectional area having (i) a distal end connected to the end of thetip at a tip point of connection, (ii) a proximal end opposite thedistal end, and (iii) a longitudinal length with a radius a from thedistal end to the proximal end; wherein the ratio of a_(t)/a is1.05-1.4, the wedge angle γ is 20-55, and the pull-off stress at thefiber surface is 0.7-0.9; and wherein the structure has a dimensionlessvalue x in the range of 1-100, with${\chi \equiv \frac{\sigma_{o}^{2}{a\left( {1 - v^{2}} \right)}}{2\pi \; {Ew}_{adh}}} = \frac{\sigma_{o}{a\left( {1 - v^{2}} \right)}}{2\pi \; E\; \delta_{c}}$wherein a=the radius of the fiber stem; E=elastic modulus; ν=thePoisson's ratio of the fiber; σ_(o)=intrinsic adhesive strength;δ_(c)=critical separation distance; and ω_(adh)=work of adhesion, orσ_(o)δ_(c).
 15. The dry adhesive fiber structure according to claim 14,wherein ratio of a_(t)/a is 1.1-1.2, the wedge angle γ is 40-50, and thenormalized pull-off stress at the fiber surface is 0.8-0.9.
 16. The dryadhesive fiber structure according to claim 14, wherein ratio of a_(t)/ais 1.1-1.2, the wedge angle γ is 45, and the normalized pull-off stressat the fiber surface at 0.9.
 17. The dry adhesive fiber according toclaim 14, wherein the tip edge formed where the tip surface meets thetip inclined sides has a radius of curvature of less than 10 μm forwetted polymer materials.
 18. The dry adhesive fiber according to claim14, wherein the tip edge formed where the tip surface meets the tipinclined sides has maximum sharpness for the fiber material andmanufacturing process used to create the fiber.
 19. The dry adhesivefiber according to claim 14, wherein the fiber length varies from 0.5 nmto 500 μm.
 20. The dry adhesive fiber according to claim 14, wherein theshape of the inclined sides on the underside of the mushroom tip isselected from the group consisting of linear, convex, and concave shapeswith respect to the stem axial direction and tip surface.